I would hazard a guess that a Sears air compressor running from 1 phase power, assuming 120V, is most certainly a capacitor start motor and could most definately cause problems on the switch mode power supply. If you cannot separate the power sources, install a line conditioner in front of the power supply. Ferroresonent are good for this application, small, inexpensive and reliable.
Air compressors tend to turn on and off quite frequently and the power quality during starting is extremely poor until the centrifugal switch on the motor takes out the caps. The problems are very breif, but the comopressor cycles very rapidly. If you dont kow, look for a "bump" on the side of the compressor motor. That is a tell tale sign of a capacitor start. The only other type of 1 phase motor would be a "split phase" starting version, no longer commonly used in machinery of this type. Some small comnpressors are using Universal motors (the type used on your power drill for instance) even though they are not using variable speed capability, but they are typically limited to fractional HP.
Actually, in reviewing you original post I see that this is a $50 compressor, so that may in deed be the case. A universal motor usesa rectifier and brushes going to DC windings, very poor power quality and very likely to do damage to your PS. Invest in a line conditioner. That which does not kill me, makes me stronger... and pissed off!